We discussed the two anomalies in the example of non-relativistic chiral Weyl superfluids and superconductors, which in the low-energy approximation have chiral (Majorana-)Weyl fermions on anisotropic are Newton-Cartan geometries. Along the anisotropy direction and linear approximation, the geometry is effectively relativistic and the relativistic CVE and CTE results can be applied. The CVE is induced by the rotating normal component, whereas the torsionful tetrad arises from the superfluid component with \({v}_s\) . While CTE here superficially originates from condensate vorticity, we note that, by Mermin-Ho relations, \({v}_s\)  is related in hydrodynamics to the order parameter frame fields \(e^a_i = \lbrace \hat{{m}}, \hat{{n}}, \hat{{l}}\rbrace \) , of which only the \(\hat{{l}}\)  is a well-posed hydrodynamic variable in addition to, \(\mu , T, \rho _s, \rho _n, {v}_s, {v}_n\)  (spin is neglected). The perpendicular \(\hat{{m}}, \hat{{n}}\) -components contribute to \({v}_s\)  and represent gauge degrees of freedom in terms of the combined orbital-phase symmetry of the superfluid (superconductor) [1]}. In the absence of relativistic systems with torsion, this comparison of the two anomalies with independent experimentally available sources is the best we can do. To see the relation of the two anomalies, we had to “bias" \({v}_s\)  with \({v}_n\)  and go higher order in gradients as compared to [2]}. In this case, the absence of chiral currents in equilibrium is obtained by the cancellation of the two anomaly currents, spatially averaged over the (unit cell) of the vortex lattice. To maintain the validity of the relativistic approximation with spin-1/2 chiral fermions, we considered the case where \(\hat{{l}}\)  is constant and along rotation axis \({\Omega }\) , corresponding to pure phase vortices [3]}.
